I have very basic question about reverse mode algorithm calculation. It is well-known that we calculate derivatives according to chain rule, using sub expressions. I was trying to find kind of proof, that it is necessary to use $\textbf {addition}$ for the computation of variable, which is an input for 2 sub expressions.
For example for this graph we calculate $b=w_2$ through $c=w_3, d=w_4 $ by $\textbf {addition}$ -
$\bar w_2 = \bar w_3 \frac {dw_3}{dw_2} \textbf {+} \bar w_4 \frac {dw_4}{dw_2}$
If you know about the nature of this addition or any materials where I can find the proof please write it. I will really appreciate it.